Many answers are possible. The point of this note is to refute one
particular common answer, which is that the whole thing is just
meaningless.

This view is espoused by many people who, it seems, ought to know
better. There are two problems with this view.

The first problem is that it involves a theory of meaning that appears to
have nothing whatsoever to do with pragmatics. You can certainly
say that something is meaningless, but that doesn't make it so.
I can claim all I want to that "jqgc ihzu kenwgeihjmbyfvnlufoxvjc sndaye"
is a meaningful utterance, but that does not avail me much, since
nobody can understand it. And conversely, I can say as loudly and as
often as I want to that the utterance "Snow is white" is meaningless,
but that doesn't make it so; the utterance still means that snow is
white, at least to some people in some contexts.

Similarly, asserting that the sentences are meaningless is all very
well, but the evidence is against this assertion. The meaning of the
utterance "sentence 2 is false" seems quite plain, and so does the
meaning of the utterance "sentence 1 is false". A theory of meaning
in which these simple and plain-seeming sentences are actually
meaningless would seem to be at odds with the evidence: People do
believe they understand them, do ascribe meaning to them, and, for
the most part, agree on what the meaning is. Saying that "snow is
white" is meaningless, contrary to the fact that many people agree
that it means that snow is white, is foolish; saying that the example
sentences above are meaningless is similarly foolish.

I have heard people argue that although the sentences are individually
meaningful, they are meaningless in conjunction. This position is
even more problematic. Let us refer to a person who holds this
position as P.
Suppose
sentence 1 is presented to you in isolation. You think you understand
its meaning, and since P agrees that it is meaningful, he
presumably would agree that you do. But then, a week later, someone
presents you with sentence 2; according to P's theory, sentence
1 now becomes meaningless. It was meaningful on February 1, but not
on February 8, even though the speaker and the listener both think it
is meaningful and both have the same idea of what it means. But
according to P, as midnight of February 8, they are suddenly
mistaken.

The second problem with the notion that the sentences are meaningless
comes when you ask what makes them meaningless, and how one can
distinguish meaningful sentences from sentences like these that are
apparently meaningful but (according to the theory) actually
meaningless.

The answer is usually something along the lines that sentences that
contain self-reference are meaningless. This answer is totally
inadequate, as has been demonstrated many times by many people,
notably W.V.O. Quine. In the example above, the self-reference
objection is refuted simply by observing that neither sentence is
self-referent. One might try to construct an argument about
reference loops, or something of the sort, but none of this will
avail, because of Quine's example:

"is false when appended to a quoted version of itself."
is false when appended to a quoted version of itself.
This is a perfectly well-formed, grammatical sentence (of the form
"x is false when appended to a quoted version of itself".) It
is not immediately self-referent, and there is no "reference loop"; it
merely describes the result of a certain operation. In this way, it
is analogous to sentences like this one:

"snow is white" is false when you change "is" to "is not".

Or similarly:

If a sentence is false, then its negation is true.

Nevertheless, Quine's sentence is an antinomy of the same sort as the
example sentences at the top of the article.

But all of this is peripheral to the main problem with the argument
that sentences that contain self-reference are meaningless. The main
problem with this argument is that it cannot be true. The
sentence "sentences that contain self-reference are meaningless" is
itself a sentence, and therefore refers to itself, and is therefore
meaningless under its own theory. If the assertion is true, then the
sentence asserting it is meaningless under the assertion itself; the
theory deconstructs itself. So anyone espousing this theory has
clearly not thought through the consequences. (Graham Priest says that
people advancing this theory are subject to a devastating ad
hominem attack. He doesn't give it specifically, but many such
come to mind.)

In fact, the self-reference-implies-meaninglessness theory obliterates
not only itself, but almost all useful statements of logic. Consider
for example "The negation of a true sentence is false and the negation
of a false sentence is true." This sentence, or a variation of it, is
probably found in every logic textbook ever written. Such a sentence
refers to itself, and so, in the
self-reference-implies-meaninglessness theory, is meaningless. So too
with most of the other substantive assertions of our logic textbooks,
which are principally composed of such self-referent sentences about
properties of sentences; so much for logic.

The problems with ascribing meaninglessness to self-referent sentences
run deeper still. If a sentence is meaningless, it cannot be
self-referent, because, being meaningless, it cannot refer to anything
at all. Is "jqgc ihzu kenwgeihjmbyfvnlufoxvjc sndaye" self-referent?
No, because it is meaningless. In order to conclude that it was
self-referent, we would have to understand it well enough to ascribe a
meaning to it, and this would prove that it was meaningful.

So the position that the example sentences 1 and 2 are "meaningless"
has no logical or pragmatic validity at all; it is totally
indefensible. It is the philosophical equivalent of putting one's
fingers in one's ears and shouting "LA LA LA I CAN'T HEAR YOU!"